Twice or three times, I've submitted a solution, only to receive a message stating that the game was started by a logged in user and that I must log back in (and complete the entire puzzle again!) to submit it.

I have had that happen, but realized that I neglected to click "log in" on the second screen, just went ahead and clicked on "New Game". Make sure that it says your screen name before you start a new game (puzzle).

Yep, I've had it but not for a while. I think it's got something to do with timing out because that's when it used to happen to me. Now, when I see a 17% or 25%, I immediately click 'new puzzle' up on the top and avoid the crazy challenging ones.
My last solution when it took me a long time just staring and trying to figure it out, I hit 'print screen' on the keyboard, opened up MS paint, clicked the pull down menu 'edit' and clicked paste. This way if I ended up being logged out, alteast I could get my answers in without another 15mins.
Another way that I haven't tried may be to keep another tab open, and make sure your logged in there before clicking submit answers.